H. Algebra 2
10.1 Inverses of Simple Quadratic and Cubic Functions
Date: ___________
Explore. Finding the Inverse of a Many-to-One Function
The function 𝑓(𝑥) is defined by the following ordered pairs: (−2, 4), (−1, 2), (0, 0), and (2, 4).
A. Find the inverse of 𝑓(𝑥), 𝑓 −1 (𝑥), by reversing the coordinates in the ordered pairs.
B. Is the inverse also a function?
C. If necessary, restrict the domain of 𝑓(𝑥) such that the inverse, 𝑓 −1 (𝑥), is also a function.
D. With the restricted domain of 𝑓(𝑥), what ordered pairs will define the inverse function 𝑓 −1 (𝑥)?
Finding and Graphing the Inverse of a Simple Quadratic Function
Learning Target A: I can find and graph the inverse of a simple quadratic function.
The function 𝑓(𝑥) = 𝑥 2 is a many-to-one function, so its domain must be restricted in order to find its
inverse function.
𝑓(𝑥) = 𝑥 2
(-3, 9)
(3, 9)
If its domain is restricted to
𝑥 ≥ 0, then its inverse
function is 𝑓 −1 (𝑥) = √𝑥.
If its domain is restricted to
𝑥 ≤ 0, then its inverse
function is 𝑓 −1 (𝑥) = −√𝑥.
(-2, 4)
(-1, 1)
(2, 4)
(1, 1)
(0, 0)
The parent square root function is 𝒈(𝒙) = √𝒙.
What is the domain of the parent square root function?
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Restrict the domain of each quadratic function and find its inverse. Confirm the inverse
relationship using composition. Graph the function and its inverse.
A. 𝑓(𝑥) = 0.5𝑥 2
B. 𝑓(𝑥) = 𝑥 2 − 7
C. 𝑓(𝑥) = 3𝑥 2
Finding the Inverse of a Quadratic Model
In many instances, quadratic functions are used to model real-world applications. It is often useful to
find and interpret the inverse of a quadratic model. Note that when working with real-world
applications, it is more useful to use the notation 𝑥(𝑦) for the inverse of 𝑦(𝑥) instead of using the
notation 𝑦 −1 (𝑥).
Learning Target B: I can write and use inverse quadratic models to evaluate real-life situations.
Find the inverse of each of the quadratic functions. Use the inverse to solve the application.
A. The function 𝑑(𝑡) = 16𝑡 2 gives the distance, d, in feet that a dropped object falls in t seconds.
Write the inverse function 𝑡(𝑑) to find the time t in seconds it takes for an object to fall a distance of d
feet. Then estimate how many seconds it will take a penny to drop 48 feet.
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B. The function 𝐸(𝑣) = 4𝑣 2 gives the kinetic energy E in Joules of an 8-kg object that is traveling at a
velocity of v meters per second. Write the inverse function 𝑣(𝐸) to find the velocity v in meters per
second required for an 8-kg object to have a kinetic energy of E Joules. Then estimate the velocity
required for an 8-kg object to have a kinetic energy of 60 Joules.
C. The function 𝐴(𝑟) = 𝜋𝑟 2 gives the area of a circular object with respect to its radius r. Write the
inverse function 𝑟(𝐴) to find the radius r required for an area of A. Then estimate the radius of a
circular object that has an area of 40 cm2.
Finding and Graphing the Inverse of a Simple Cubic Function
Learning Target C: I can find and graph the inverse of a simple cubic function.
Note that the function 𝑓(𝑥) = 𝑥 3 is a one-to-one function, so its domain does not need to be
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restricted in order to find its inverse function. The inverse of 𝑓(𝑥) = 𝑥 3 is 𝑓 −1 (𝑥) = √𝑥.
𝟑
Parent Cube Root Function: 𝒈(𝒙) = √𝒙
Find the inverse of each cubic function. Confirm the inverse relationship using composition.
Graph the function and its inverse.
A. 𝑓(𝑥) = 0.5𝑥 3
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B. 𝑓(𝑥) = 𝑥 3 − 9
C. 𝑓(𝑥) = 2𝑥 3
Finding the Inverse of a Cubic Model
Learning Target D: I can write and use inverse cubic models to evaluate real-life situations.
Similar to quadratic models, cubic functions can be used to model many real-world applications.
Find the inverse of each of the following cubic functions.
A. The function 𝑚(𝐿) = 0.00001𝐿3 gives the mass m in kilograms of a red snapper of length L
centimeters. Find the inverse function 𝐿(𝑚) to find the length L in centimeters of a red snapper that
has a mass of m kilograms.
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B. The function 𝐴(𝑟) = 3 𝜋𝑟 3 gives the surface area A of a sphere with radius r. Find the inverse
function 𝑟(𝐴) to find the radius r of a sphere with surface area A.
C. The function 𝑚(𝑟) =
44
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𝜋𝑟 3 gives the mass in grams of a spherical lead ball with a radius of r
centimeters. Find the inverse function 𝑟(𝑚) to find the radius r of a lead sphere with mass m.
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